A304678 -id:A304678 - cp's OEIS frontend

A329131 Numbers whose prime signature is a Lyndon word.

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 50, 53, 54, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 89, 97, 98, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 147, 149, 150, 151, 157, 162, 163, 167

Offset: 1

Gus Wiseman, Nov 06 2019

nonn

First differs from A133811 in having 50.
A Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations.
A number's prime signature is the sequence of positive exponents in its prime factorization.

			The prime signature of 30870 is (1,2,1,3), which is a Lyndon word, so 30870 is in the sequence.
The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   18: {1,2,2}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}

Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose prime signature is a necklace are A329138.
Numbers whose prime signature is aperiodic are A329139.
Lyndon compositions are A059966.
Prime signature is A124010.
Cf. A000031, A000740, A000961, A001037, A025487, A027375, A097318, A112798, A118914, A304678, A318731, A329140, A329142.

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
Select[Range[2,100],lynQ[Last/@FactorInteger[#]]&]

Intersection of A329138 and A329139.

A097318 Numbers with more than one prime factor and, in the ordered factorization, the exponent never increases when read from left to right.

Original entry on oeis.org

6, 10, 12, 14, 15, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 99, 100, 102, 104, 105, 106, 110, 111, 112, 114

Offset: 1

Ralf Stephan, Aug 04 2004

nonn

If n = Product_{k=1..m} p(k)^e(k), then m > 1, e(1) >= e(2) >= ... >= e(m).

These are numbers whose ordered prime signature is weakly decreasing. Weakly increasing is A304678. Ordered prime signature is A124010. - Gus Wiseman, Nov 10 2019

			60 is 2^2*3^1*5^1, A001221(60)=3 and 2>=1>=1, so 60 is in sequence.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
S. Ramanujan, Asymptotic formulas for the distribution of integers of various types, Proc. London Math. Soc. 2, 16 (1917), 112-132.

Subset of A024619. Cf. A097319, A097320, A230766.

Cf. A001222, A025487, A118914, A124010, A304678, A329138, A329142.

q:= n-> (l-> (t-> t>1 and andmap(i-> l[i, 2]>=l[i+1, 2],
        [$1..t-1]))(nops(l)))(sort(ifactors(n)[2])):
select(q, [$1..120])[];  # Alois P. Heinz, Nov 11 2019

fQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Length[f] > 1 && Max[Differences[f]] <= 0]; Select[Range[2, 200], fQ] (* T. D. Noe, Nov 04 2013 *)

for(n=1, 130, F=factor(n); t=0; s=matsize(F)[1]; if(s>1, for(k=1, s-1, if(F[k, 2]

A304686 Numbers with strictly decreasing prime multiplicities.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 52, 53, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 99, 101, 103, 104, 107, 109, 112, 113, 116, 117, 121

Offset: 1

Gus Wiseman, May 16 2018

nonn

			10 = 2*5 has prime multiplicities (1,1) so is not in the sequence.
20 = 2*2*5 has prime multiplicities (2,1) so is in the sequence
90 = 2*3*3*5 has prime multiplicities (1,2,1) so is not in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A001597, A005117, A055932, A071365, A112769, A130091, A133808, A133811, A181819, A242031, A304678, A304679, A283050.

Select[Range[200],Greater@@FactorInteger[#][[All,2]]&]

isok(n) = my(vm = factor(n)[,2]); vm == vecsort(vm,,4) && (#vm == #Set(vm)); \\ Michel Marcus, May 17 2018

list(lim)=my(v=List()); forfactored(n=1,lim\1, if(n[2][,2]==vecsort(n[2][,2],,8), listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Oct 28 2021

a(n) ~ n log n. - Charles R Greathouse IV, Oct 28 2021

A329139 Numbers whose prime signature is an aperiodic word.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 101, 103, 104

Offset: 1

Gus Wiseman, Nov 09 2019

nonn

First differs from A319161 in having 1260 = 2*2 * 3^2 * 5^1 * 7^1. First differs from A325370 in having 420 = 2^2 * 3^1 * 5^1 * 7^1.
A number's prime signature (A124010) is the sequence of positive exponents in its prime factorization.
A sequence is aperiodic if its cyclic rotations are all different.

			The sequence of terms together with their prime signatures begins:
   1: ()
   2: (1)
   3: (1)
   4: (2)
   5: (1)
   7: (1)
   8: (3)
   9: (2)
  11: (1)
  12: (2,1)
  13: (1)
  16: (4)
  17: (1)
  18: (1,2)
  19: (1)
  20: (2,1)
  23: (1)
  24: (3,1)
  25: (2)
  27: (3)

Complement of A329140.
Aperiodic compositions are A000740.
Aperiodic binary words are A027375.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose prime signature is a Lyndon word are A329131.
Numbers whose prime signature is a necklace are A329138.
Cf. A025487, A097318, A112798, A124010, A178472, A181819, A304678, A329133, A329135, A329136, A329137, A329142.

aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Select[Range[100],aperQ[Last/@FactorInteger[#]]&]

A383100 Numbers whose prime indices have no permutation with all equal run-sums.

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 42, 44, 45, 46, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108

Offset: 1

Gus Wiseman, Apr 20 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The prime indices of 18 are {1,2,2}, with permutations (1,2,2), (2,1,2), (2,2,1), with run sums (1,4), (2,1,2), (4,1) respectively, so 18 is in the sequence.
The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   46: {1,9}
   50: {1,3,3}

For distinct instead of equal run-sums we appear to have A381636, counted by A381717.
For run-lengths instead of sums we have A382879, counted by complement of A383013.
These are the positions of 0 in A382877.
For more than one choice we have A383015.
The complement is A383110, counted by A383098.
Partitions of this type are counted by A383096.
For a unique choice we have A383099, counted by A383095.
A056239 adds up prime indices, row sums of A112798.
A304442 counts partitions with equal run-sums, ranks A353833.
A353851 counts compositions with equal run-sums, ranks A353848.
Cf. A047966, A072774, A130091, A242031, A304686, A304678, A334965.
Cf. A351294, A351295, A353832, A353837, A353838, A354584, A381871, A382857, A382876, A383094, A383097.

Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Total/@Split[#]&]]==0&]

A329138 Numbers whose prime signature is a necklace.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83

Offset: 1

Gus Wiseman, Nov 09 2019

nonn

First differs from A304678 in having 1350 = 2^1 * 3^3 * 5^2. First differs from A316529 in having 150 = 2^1 * 3^1 * 5^2.
A number's prime signature (A124010) is the sequence of positive exponents in its prime factorization.
A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.

			The sequence of terms together with their prime signatures begins:
   2: (1)
   3: (1)
   4: (2)
   5: (1)
   6: (1,1)
   7: (1)
   8: (3)
   9: (2)
  10: (1,1)
  11: (1)
  13: (1)
  14: (1,1)
  15: (1,1)
  16: (4)
  17: (1)
  18: (1,2)
  19: (1)
  21: (1,1)
  22: (1,1)

Complement of A329142.
Binary necklaces are A000031.
Necklace compositions are A008965.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose prime signature is a Lyndon word are A329131.
Numbers whose prime signature is aperiodic are A329139.
Cf. A001037, A025487, A056239, A097318, A112798, A118914, A124010, A181819, A304678, A328596, A329140.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[2,100],neckQ[Last/@FactorInteger[#]]&]

A353507 Product of multiplicities of the prime exponents (signature) of n; a(1) = 0.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 2, 2, 2, 2

Offset: 1

Gus Wiseman, May 19 2022

nonn

Warning: If the prime multiplicities of n are a multiset y, this sequence gives the product of multiplicities in y, not the product of y.
Differs from A351946 at A351946(1260) = 4, a(1260) = 2.
Differs from A327500 at A327500(450) = 3, a(450) = 2.
We set a(1) = 0 so that the positions of first appearances are the primorials A002110.
Also the product of the prime metasignature of n (row n of A238747).

			The prime signature of 13860 is (2,2,1,1,1), with multiplicities (2,3), so a(13860) = 6.

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of first appearances are A002110.
The prime indices themselves have product A003963, counted by A339095.
The prime signature itself has product A005361, counted by A266477.
A001222 counts prime factors with multiplicity, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A071625 counts distinct prime exponents (third omega).
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, sorted A353742.
A323022 gives fourth omega.
Cf. A085629, A097318, A182850, A304678, A353394, A353399, A353500, A353503.

f:= proc(n) local M,s;
  M:= ifactors(n)[2][..,2];
  mul(numboccur(s,M),s=convert(M,set));
end proc:
f(1):= 0:
map(f, [$1..100]); # Robert Israel, May 19 2023

Table[If[n==1,0,Times@@Length/@Split[Sort[Last/@FactorInteger[n]]]],{n,100}]
Join[{0},Table[Times@@(Length/@Split[FactorInteger[n][[;;,2]]]),{n,2,100}]] (* Harvey P. Dale, Oct 20 2024 *)

from math import prod
from itertools import groupby
from sympy import factorint
def A353507(n): return 0 if n == 1 else prod(len(list(g)) for k, g in groupby(factorint(n).values())) # Chai Wah Wu, May 20 2022

a(n) = A005361(A181819(n)) = A003963(A181819(A181819(n))).

A329140 Numbers whose prime signature is a periodic word.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154

Offset: 1

Gus Wiseman, Nov 09 2019

nonn

First differs from A182853 in having 2100 = 2^2 * 3^1 * 5^2 * 7^1.
A number's prime signature (A124010) is the sequence of positive exponents in its prime factorization.
A sequence is aperiodic if its cyclic rotations are all different.

			The sequence of terms together with their prime signatures begins:
   6: (1,1)
  10: (1,1)
  14: (1,1)
  15: (1,1)
  21: (1,1)
  22: (1,1)
  26: (1,1)
  30: (1,1,1)
  33: (1,1)
  34: (1,1)
  35: (1,1)
  36: (2,2)
  38: (1,1)
  39: (1,1)
  42: (1,1,1)
  46: (1,1)
  51: (1,1)
  55: (1,1)
  57: (1,1)
  58: (1,1)

Complement of A329139.
Periodic compositions are A178472.
Periodic binary words are A152061.
Numbers whose binary expansion is periodic are A121016.
Numbers whose prime signature is a Lyndon word are A329131.
Numbers whose prime signature is a necklace are A329138.
Cf. A025487, A056239, A097318, A112798, A118914, A124010, A181819, A304678, A329132 A329134, A329143, A329144.

aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Select[Range[100],!aperQ[Last/@FactorInteger[#]]&]

A332836 Number of compositions of n whose run-lengths are weakly increasing.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 24, 40, 73, 128, 230, 399, 712, 1241, 2192, 3833, 6746, 11792, 20711, 36230, 63532, 111163, 194782, 340859, 596961, 1044748, 1829241, 3201427, 5604504, 9808976, 17170112, 30051470, 52601074, 92063629, 161140256, 282033124, 493637137, 863982135, 1512197655

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

Also compositions whose run-lengths are weakly decreasing.

			The a(0) = 1 through a(5) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (121)   (41)
                        (211)   (122)
                        (1111)  (131)
                                (212)
                                (311)
                                (1211)
                                (2111)
                                (11111)
For example, the composition (2,3,2,2,1,1,2,2,2) has run-lengths (1,1,2,2,3) so is counted under a(17).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for the compositions themselves (not run-lengths) is A000041.
The case of partitions is A100883.
The case of unsorted prime signature is A304678, with dual A242031.
Permitting the run-lengths to be weakly decreasing also gives A332835.
The complement is counted by A332871.
Unimodal compositions are A001523.
Compositions that are not unimodal are A115981.
Compositions with equal run-lengths are A329738.
Compositions whose run-lengths are unimodal are A332726.
Cf. A001462, A072704, A072706, A100882, A181819, A329744, A329766, A332641, A332727, A332745, A332833, A332834.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],LessEqual@@Length/@Split[#]&]],{n,0,10}]

step(M, m)={my(n=matsize(M)[1]); for(p=m+1, n, my(v=vector((p-1)\m, i, M[p-i*m,i]), s=vecsum(v)); M[p,]+=vector(#M,i,s-if(i<=#v, v[i]))); M}
seq(n)={my(M=matrix(n+1, n, i, j, i==1)); for(m=1, n, M=step(M, m)); M[1,n]=0; vector(n+1, i, vecsum(M[i,]))/(n-1)} \\ Andrew Howroyd, Dec 31 2020

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

A353742 Sorted prime metasignature of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 3, 1, 1

Offset: 1

Gus Wiseman, May 20 2022

The prime metasignature counts the multiplicities of each value in the prime signature of n. For example, 2520 has prime indices {1,1,1,2,2,3,4}, sorted prime signature {1,1,2,3}, and sorted prime metasignature {1,1,2}.

			The prime indices, sorted prime signatures, and sorted prime metasignatures of selected n:
      n = 1: {}             -> {}         -> {}
      n = 2: {1}            -> {1}        -> {1}
      n = 6: {1,2}          -> {1,1}      -> {2}
     n = 12: {1,1,2}        -> {1,2}      -> {1,1}
     n = 30: {1,2,3}        -> {1,1,1}    -> {3}
     n = 60: {1,1,2,3}      -> {1,1,2}    -> {1,2}
    n = 210: {1,2,3,4}      -> {1,1,1,1}  -> {4}
    n = 360: {1,1,1,2,2,3}  -> {1,2,3}    -> {1,1,1}

Row-sums are A001221.
Row-lengths are A071625.
Positions of first appearances are A182863.
This is the sorted version of A238747.
Row-products are A353507.
A001222 counts prime factors with multiplicity.
A003963 gives product of prime indices.
A005361 gives product of prime signature, firsts A353500 (sorted A085629).
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with strict signature, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
Cf. A000040, A000720, A070175, A097318, A182850, A304678, A325238, A353503.

Join@@Table[Sort[Length/@Split[Sort[Last/@If[n==1,{},FactorInteger[n]]]]],{n,100}]

cp's OEIS Frontend

A329131 Numbers whose prime signature is a Lyndon word.

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A097318 Numbers with more than one prime factor and, in the ordered factorization, the exponent never increases when read from left to right.

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A304686 Numbers with strictly decreasing prime multiplicities.

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A329139 Numbers whose prime signature is an aperiodic word.

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A383100 Numbers whose prime indices have no permutation with all equal run-sums.

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A329138 Numbers whose prime signature is a necklace.

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A353507 Product of multiplicities of the prime exponents (signature) of n; a(1) = 0.

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A329140 Numbers whose prime signature is a periodic word.

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